Theorem nfald 1813

Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)

Hypotheses

Ref	Expression
nfald.1	⊢ Ⅎ𝑦𝜑
nfald.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfald	⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Proof of Theorem nfald

Step	Hyp	Ref	Expression
1		nfald.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1572	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		nfald.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	2, 3	alrimih 1522	. 2 ⊢ (𝜑 → ∀𝑦Ⅎ𝑥𝜓)
5		nfnf1 1597	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜓
6	5	nfal 1629	. . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜓
7		hba1 1593	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → ∀𝑦∀𝑦Ⅎ𝑥𝜓)
8		sp 1564	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
9	8	nfrd 1573	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
10	7, 9	hbald 1544	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
11	6, 10	nfd 1576	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜓 → Ⅎ𝑥∀𝑦𝜓)
12	4, 11	syl 14	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1400  Ⅎwnf 1513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  dvelimALT  2070  dvelimfv  2071  nfeudv  2101  nfeqd  2407  nfraldw  2582  nfraldxy  2583  nfiotadw  5335  nfixpxy  6989  bdsepnft  16827